Zeno’s Paradoxes -The Stationary Arrow

Fire an arrow from your bow, and as you watch it fly through the air there is one thing you can be certain of – that arrow is moving. Right? Wrong, at least according to 5^th century BCE philosopher, and student/friend of Parmenides (yes, that Parmenides, the one who claimed there is only one thing (which he called being), and that one thing is unchanging and permanent), Zeno of Elea, who argued that the apparent motion of your arrow is an illusion. There have been a number of attempts to resolve this paradox over the centuries, until now, it is claimed by mathematicians, physicists, and even a few philosophers that Leibniz and Newton solved it with the invention of calculus. We will see that this claim is totally false, before looking at a resolution that actually works as expressed in slightly different form by three French philosophers; Jean-Paul Sartre, Maurice Merleau-Ponty, and Henri Bergson, the last of whom influenced the first two and, in my opinion, has the most complete response to Zeno.

The Paradox

So, you’ve fired your arrow, and you’re watching it move through the air, quite satisfied that you have refuted Zeno. The arrow moves; by which we mean, it undergoes some change, and in the case of motion, what changes in an object is the position it occupies. It first occupies position A, then position B, then C, and so on, until it reaches its destination at, say, point Z. Fine. However, if we isolate a single instant during the arrow’s flight (say instant 2, when the arrow is at point B), we notice that at that instant the arrow is completely at rest. It can only occupy two successive positions if we grant it two instants, which means that at any given, single instant, the arrow is at rest at the corresponding location. Since we can say exactly the same thing for every single instant in its flight, we are forced to say the arrow is motionless during the whole time that it is moving. Hence, the paradox.

The Infinitesimal Calculus

Zeno’s paradox turns on motion, which we can measure with speed; i.e. distance travelled divided by time. This approach is fine if we want to calculate the average speed of an object; i.e. speed over a length of time, but Zeno’s paradox concerned, not lengths of time, but a single instant, in which he claimed the arrow was completely at rest; i.e. that it had no speed. If physics/mathematics can show that objects have speed at a single instant, this will absolutely refute Zeno. (It is easier to see how the calculus works here if we have an accelerating or decelerating object, so I will consider the arrow at point B in which it is in the first part of its trajectory; hence accelerating. I will also assume that the points B and C represent the position of the arrow 2 and 3 seconds after the arrow has left the bow.)

So, if we want to know the average speed while the arrow was travelling between B and C, we can simply calculate the distance travelled divided by the time it took to travel that distance. But since the arrow was accelerating between B and C, this won’t tell us the instantaneous speed of the arrow at point B. What we can do then is calculate the average speed of the arrow during the time from point B to midway between points B and C. (At this point, it’s easier to switch to numbers) This gives us the average speed of the arrow between 2 and 2.5 seconds after leaving the bow. If we keep doing this for successively smaller lengths of time (i.e. taking the average speed between 2 and 2.25 seconds, then 2 and 2.125 seconds, etc.), we will discover that the average speeds we get, instead of just getting smaller and smaller, actually begin to approach a limit, and this is the key. So, our first measurement of average speed might yield 15 m/s (between 2 and 3 seconds), then the next one 14.75 m/s (between 2 and 2.5 seconds), then 14.52 (between 2 and 2.25 seconds), then 14.33, then 14.12, 14.012, etc. As you can see, the values approach 14 m/s but never get there. The limit is 14 m/s, which we never actually calculate, but calculus allows us to unambiguously determine. Voila! Speed at an instant.

This is all well and good. The math is as sound and certain as it gets… but does it refute Zeno? Zeno’s claim was that motion is an illusion. He ‘proved’ this by noting that an arrow at a single instant in time (i.e. when Δt = 0) isn’t moving because, by definition, it has no speed. Calculus shows that we can calculate speed at a single instant in time, but it does this by assuming that the arrow had average speeds throughout the infinite series leading up to (but never reaching) point B; that is to say, by already assuming that the arrow is moving. The mathematics (as ingenious as it is) focuses on the calculation of speed at an instant, but forgets that the whole point of the exercise is to prove motion. If you assume the arrow is moving, then it obviously follows that it also has a speed at any particular instant because it is merely passing through the corresponding location, but if you assume the arrow is moving, there is no reason to bother calculating instantaneous speed because the whole point of the exercise was to prove that motion is possible. If, on the other hand, motion isn’t a priori assumed, no amount of calculus will get you a speed; instantaneous or average.

Not only can mathematics not resolve Zeno’s paradox, it is, in fact, the reason why it is a paradox in the first place. Zeno’s paradox is only a paradox because it takes things that are whole and continuous; motion and time, and reduces them to granular, discrete quantities. Another way to say this is that Zeno’s paradoxes arise from attempts to mathematise what is fundamentally non-mathematical. If Zeno used a mathematical way of thinking to generate his paradox, applying more maths to it isn’t going to get us out. What we have to do is de-mathematise the whole thing.

Jean-Paul Sartre

The way Sartre approaches Zeno’s paradox is through a consideration of ‘being,’ or ontology. In Sartre, ‘being’ basically amounts to the most general category under which we can group everything. It turns out that we need two categories of being to describe life as we know it – being-in-itself, and being-for-itself. The being of the arrow belongs to the former, and this category contains everything that isn’t consciousness; everything that simply is what it is (unlike consciousness, which belongs to the latter, and for which negation, lack, and possibility are all central aspects).

Zeno is right that the arrow at rest at point B, and the same arrow passing through point B, are identical because, as being-in-itself, the arrow lacks the capacity to reach beyond itself to something it currently isn’t, but could be, to lean forward into a new position, as it were. The arrow is fully self-contained as what it is, wherever it happens to be. We also know that the arrow’s “quiddity” (its essence; what makes it this arrow) can’t change during its journey, or when it arrives at its destination, it will be a different arrow from the one that began the motion. Sartre’s goal then is to identify what the being of a moving body is such that its quiddity remains unchanged while its being as an object in motion is distinct from its being when it is at rest.

The location of a thing is explained by Sartre in terms of spatiality, which is that thing’s relation of exteriority to the background of objects on which it appears. This relation of exteriority will be stable and unchanging for an object at rest, but for our arrow in motion, the relation it has with the background upon which it appears (for a viewing subject) is in constant flux. This means the object has, in a way, ‘broken free’ of the original, stable external relation it had when it was at rest. Because the arrow’s being as an object in space is, at least in part, defined by that relation of exteriority, it is therefore, in a sense, ‘outside’ of this relation and therefore exterior to itself. In other words, the arrow can no longer be said to be purely what it is, fully contained in-itself at point B. Yes, it is at point B, but it is not ‘established’ there the way it would be if it were at rest. Instead, it is now reaching beyond itself, or leaning forward into a new location. In a manner of speaking (although not literally, of course), the arrow wants to move to point C. It’s almost like the arrow is trying to outrun its being-in-itself (grounded in the spatiality defined by that unchanging relation of exteriority). It is this change in thinking that secures the change in being Sartre was after.

The qualities of the arrow don’t change (all the things that make it what it is; it’s essence or quiddity), but its being (which we might think of as its mode of existence) changes because it is now “suspended between abolition and permanence”, fluctuating between the two. Consider motion that progresses from A to B to C. When the arrow is at B it has abolished what it was at A, but insofar as it “rediscovers” itself, or we could even say partially/temporarily ‘reunites’ with itself, at B, it annihilates that previous abolition. At C, it will do the same thing with respect to B, and so on until it finally comes to a rest, at which point its being will ‘settle’ into itself once more, and it will reaffirm its in-itself status.

This new mode of being Sartre names being-in-motion, which is different from being-in-itself because, instead of being fully ‘self-contained’ and ‘complete’ as what it is, it is actually exterior to itself. This makes it similar to being-for-itself (which I noted earlier is characterised by negation, or not being what it is, or again, leaning into future possibilities), but it never fully attains this kind of being because it is unable to relate to its own being the way consciousness does.[1]

This explanation does, I think, describe motion in such a way that Zeno’s paradox is avoided, and it does this, not via a direct assault on the paradox which would attempt to ‘prove’ motion on Zeno’s terms, which is, I think, impossible, but by proposing an alternative framework in which motion just doesn’t produce this logical inconsistency. As such, Zeno could maintain his paradox simply by denying Sartre’s ontology, but in the face of two ontologies, one of which explains the world as we perceive and live it, and the other of which asserts that what we see with our own eyes, not just isn’t happening, but can’t happen, which one are you more likely to opt for?

Maurice Merleau-Ponty

Sartre’s being-in-motion is entirely dependent on his account of spatiality, in which the arrow is situated in that relation of exteriority to the things around it, and, rather than questioning Zeno’s assertion that the arrow passes through every point on its trajectory, Sartre redefines what it means for the arrow to be in each point. Merleau-Ponty, while still ultimately making an argument from ontology, explains movement in such a way that neither of these two things hold true; i.e. he will claim that movement isn’t dependent on external relations, and that the arrow doesn’t pass through every point.

Merleau-Ponty starts by noting that the very attempt to think movement, to dissect it in order to render it understandable, unavoidably distorts it. The dissected account tells us two things. First, the moving object doesn’t change; and second, the movement doesn’t take place in the moving object itself. Instead, it manifests as a change in the relations between the object and its environment. This means that the movement and the moving object are separate, and inevitably leads to the paradox in which the ‘moving’ object does not move.

To counter this intellectualised picture of movement, Merleau-Ponty suggests we look at motion as it appears to us; i.e. without making any reductive inferences. If we pay attention to our immediate perception of the arrow in flight, the first thing we notice is that we don’t see the arrow passing through a series of intermediate positions successively. Instead, we see it in three ‘states;’ beginning, carrying out, and completing the movement. Digging a little deeper, we see that the arrow at the beginning, and the arrow at the end are the same; i.e. they have the same properties. The arrow in the middle of the movement, however; rather than having different properties, completely lacks determinate properties at all. It doesn’t have a certain length, a determinate shape, or a definite colour; the most we can say is that it is a blurry, “colored something,” too indistinct to be said to possess any actual properties. Instead of properties, Merleau-Ponty says it possesses a “style.” He captures this difference between the arrow at either end of the movement and in the middle of the movement by calling the former a movable object, and the latter, a moving object.

The moving object is actually pre-logical and non-thetic (a word which means not explicitly before one as a determinate object); i.e. it is something we ‘live,’ or ‘experience,’ rather than something we explicitly and determinately ‘know.’ The objective, analytical, knowledge of motion we came to through dissecting the whole into parts, rather than getting to the heart of motion, turns out to be a secondary, derived abstraction that fails to describe motion because it has broken it apart, and replaced it with bits and pieces (a series of arrows linked together in succession) that don’t move. In general terms, what Merleau-Ponty is doing here is trying to recapture the lived, uncertain, non-thetic field within which we actually live, and which is prior to, and embraces, the objective, thetic one we create when we seek to delineate, define, and understand things intellectually.

Motion only exists as a whole, bordered by two stationary objects. Attempting to break it up into individual instants destroys it. But isn’t this exactly what Zeno is claiming; that when we go beyond appearances, we find that the motion we thought we saw is actually an illusion? Not quite. We typically think that in objectively dissecting phenomena in this way, we are getting at the ‘truth’ of things, but that attitude is precisely what Merleau-Ponty is questioning. Disdaining subjectivity in favour of a disinterested, neutral observer, rather than getting us closer to reality, actually distorts it. This isn’t to argue that my subjective opinions or beliefs can trump objective reality; that way lies the excesses of postmodernism. The subjectivity Merleau-Ponty is talking about here goes deeper than personal opinion. Instead, he is talking about being a subject; i.e. perceiving the world from a limited, finite, human perspective; a perspective that you can’t override simply by willing something different. In short, it is the rejection of the scientific consensus view that reality or world exists in any meaningful way without a subject to engage with, or care about, it. In a world without subjects, not only would there be no motion, there wouldn’t be any arrows, and not for the trivial reason that there would be no one to make them, but for the deeper reason that there would exist no perspective in which the particles/quantum field excitations the arrow is comprised of would stand out against the surrounding particle/quantum field excitations.

Henri Bergson

The last of the three philosophers we will look at in this article, Bergson, is actually the first, in chronological terms, and it is from his refutation of Zeno’s paradox that Sartre and (especially) Merleau-Ponty derive theirs. Despite this, the differences between the three are instructive, and for reasons that will presently become clear, it is Bergson, I think, who gives the most complete and thorough critique of Zeno.

The differences amongst the three philosophers can perhaps most easily be elucidated in the focus they place on the subject. We have already seen that for Sartre the conscious subject plays a pivotal role in motion by establishing the external relations between the arrow and its surroundings. In Merleau-Ponty, the subject retains a central role although, as we saw, motion only appears for the non-thetic subject, not the thetic one; meaning it is something that, in principle, cannot be broken into parts and quantified to be made accessible to the analysing intellect. Instead, it must be lived. Bergson, as we’ve already seen in Merleau-Ponty (although I’m working backwards chronologically, remember), sees the motion of the arrow as a fundamental and irreducible whole, but he differs in two ways; first, he completely divests himself of the subject, locating motion in the arrow itself, and second, he explains why it is that we have this tendency to decompose movement (and everything else, in fact), and try to understand it as an assemblage of discrete parts.

For Bergson, change or becoming is a central and fundamental feature of the universe, and it is this independent of human consciousness. This approach distinguishes him as what we might call a ‘life philosopher’ from his two later compatriots, who are dyed in the wool phenomenologists. It also means there are no philosophical obstacles to Bergson truly placing motion in things. It isn’t just that motion is perceived as a whole by a subject (non-thetic or otherwise), it is a whole, in and of itself. Movement, like all change, all becoming, is fundamentally indecomposable. Of course, the fact that it can’t be broken apart doesn’t stop us from experiencing movement and change in our daily lives all the time; it’s just that precisely because we experience it, because we live it, we don’t know it. It remains something unquantified, non-objective; something we can only know through intuition (and with this word, Bergson doesn’t mean ‘use your feelings, man’; rather, he means non-intellectually or non-analytically); i.e. not rigorous, quantitative analysis. The trajectory, or motion, of the arrow is created in a single stroke (although over a certain extent of duration, which, by its nature, is also indivisible), and is unable to be divided precisely because it is a process, not a thing. However, once the act is completed, once the arrow reaches its destination, we are able to imagine the trajectory as a line in space, thereby turning it into a thing we are subsequently able to divide into as many immobile instants as we like, and, of course, generate paradoxes out of. (This theoretical ability to divide this quantitative derivative of true motion into an infinite number of parts is a strong hint by the way that it is an abstract figment of our imagination, not a concrete part of reality) And yet, we insist on doing just this. Why?

Our tendency to mathematise, or reduce to quantitative parts, processes that are fundamentally continuous arises from our evolutionary history. Briefly, as animals that had to act to be successful, we had to have both goals, and the means for achieving these goals worked out. This latter required us to understand the world in mechanical, causal terms so that we were able to predict and control future effects based on present states. As civilisation progressed, these mechanical models of the world became increasingly rigorous and systematic, reaching their apotheosis with mathematics. This tendency to mathematise, as science has demonstrated again and again, is of immense practical value, but a practicality that is only able to be secured at the expense of a proper grasp of reality itself. All of this has led to our current predicament in which we have created mathematical, abstract models of reality so impressive that we have forgotten they are only models, and mistaken them for reality itself.

Conclusion

Zeno’s paradoxes have fascinated and intrigued ever since he first devised them two-and-a-half thousand years ago. Despite having been thoroughly refuted by the three French philosophers we have looked at here (and no doubt by others whom I haven’t mentioned – I think Aristotle, for one, mounted a similar response by denying that time can be broken into instants), the resolution has largely gone unnoticed, even by those who are aware of them. The reason for this is, I suggest, twofold. First, as befits our science/mathematics-obsessed age, many believe the infinitesimal calculus is somehow able to overcome a paradox, which was originally generated by mathematising the unmathematisable, by applying more mathematics to it. Secondly, as Bergson noted, it is our natural instinct to mathematise reality; that is, to dismantle it into objective, quantitative terms amenable to our intellects as neat, ordered models. As long as science, which epitomises this approach, maintains its hegemony over truth, and until we make the effort to understand reality through intuition (or non-thetically), accepting the ambiguity and incompleteness that this necessarily includes, Zeno and his paradoxes will surely confound students of philosophy for another two-and-a-half thousand years.

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[1] If you’ve read Sartre, you might find it problematic that I’ve basically described being-in-itself as having within it the possibility of becoming being-in-motion; something that would necessitate a complete re-working of Sartre’s ontology. I have explained it like this for the sake of simplicity, and readers perhaps less familiar with ontology. The being of the arrow doesn’t actually change; rather, it takes on this new form of being (being-in-motion) only for an observing for-itself. In the absence of consciousness, motion would be impossible.